A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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21 views

Distribution of the supremum of a transformed Brownian Motion?

I have a stochastic process given by $z_{t}=w_{t}/\alpha\left(t\right)$ , where $w_{t}$ follows a Wiener process (a standard $\left(0,1\right)$ Brownian Motion) starting from $w_{0}=0$ , ...
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1answer
36 views

exchangeability of random variables and conditional expectation

Let $(\Omega, \mathcal{F}, \mathbb{P} )$ be a product space ${\mathbb{R}}^{\mathbb{N}}$ equipped with the product $\sigma$-algebra $\mathcal{B} ({\mathbb{R}}^{\mathbb{N}})$. Let $(X_n)_{n \geq 1}$ be ...
0
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24 views

Independent Brownian motions question

Let $B$ and $W$ be independent Brownian Motions and let $\tau$ be a stopping time of $W$. Is it true that $E[\int_0^{\tau} B_s \, dW_s] = 0\text{ ?}$ So far I have tried the following: The integral ...
1
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1answer
17 views

mean hitting time of a level and growth rate of maximum process

Let $X_t$ be the absolute value of Brownian motion starting at $0$, let $\tau_x$ be it's first hitting time of the level $x>0$, and let $M_t$ be it's running maximum up to time $t$. Suppose we knew ...
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0answers
23 views

How do you solve martingale stopping time problems?

I know this is a rather broad question, so here's a specific example: What is the distribution and expected value of the first time you get X heads during a sequence of coin tosses? Now...is there a ...
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1answer
32 views

Ito with the function containing stochastic integral

Statement of problem From Oksendal SDEs question 5.18: The geometric mean reversion process is a solution to: $$ dX_t = k (a - \log X_t) X_t dt + \sigma X_t dB_t $$ In showing that solution is ...
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0answers
30 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
0
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0answers
53 views

Ito integral solution

I need some help on how to compute the stochastic integral \begin{align} \int_{0}^{t}\frac{1}{\alpha-u}dW(u) \end{align} where $\alpha>0$.
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21 views

Optimal stopping problem

Consider the OU process: $dX_t = -X_tdt + dW_t$, $X_0 = 0$. Compute the optimal stopping time for the following problem: $$v = \sup_{\tau} E[|X_{\tau}| - \tau]$$ So far I have set $L\phi = 0$, ...
0
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1answer
20 views

Laplace transform of the autocorrelation of a wss random process

Consider a wide-sense-stationary random process $x(t)$. The autocorrelation function is $r(t-\tau):=E[x(t)x(\tau)]$. Let $S(s)$ be the Laplace-transform of $r(t)$. Can I compute $S(s)$ as ...
1
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0answers
23 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
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0answers
11 views

Entropy of non-ergodic process

Two coins have been kept in a box, One is fair while the other is biased. One coin is picked. The probability of either coin being picked is equal. The picked coin is then tossed again and again to ...
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2answers
43 views

Well-definedness of the characteristic function of a compound Poisson variable

I am reading about compound Poisson variables and cannot get through the following statement. Let $\nu$ be a non-zero finite measure on $\mathbb{R}\setminus\{0\}$. Assume that $$\int ...
0
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1answer
18 views

Invariant probability vector as a left eigenvector

What is the probability in the long run that the chain is in state 1? Solve this by directly computing the invariant probability vector as a left eigenvector. \begin{bmatrix} .4 &.2 &.4 \\ ...
3
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1answer
85 views

How to prove a stochastic process does not exist?

I'm studying the stochastic analysis material, and stuck with a problem which states below: Suppose $\{X_t, 0 \leq t \leq 1 \}$ is a real-valued stochastic process that satisfies (a) $X_s$ and ...
0
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0answers
12 views

Individual particle tracking simulation

I want to do a simulation of a stochastic system. I have 4 types of cell, each will divide or die with a certain probability. Let's say : A-> A+A A -> A+B A -> A+C B-> B+B B-> die and so on... ...
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1answer
32 views

Sum of Binomial Coefficient products

I am trying to prove that $$\sum\limits_{y=0}^d \frac{{2x \choose y} {2d-2x \choose d-y} }{2d \choose d} = x $$ So far, I have tried using induction on $d$ but I am having trouble using the ...
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1answer
17 views

$n$-step transition probability of a Markov chain

Let $(X_t)_{t\in\mathbb{N}_0}$ be a time-homogenous Markov chain over a finite state space $\left\{1,\dots,m\right\}$, so that we've got $$\Pr(X_{t+1}=j\mid X_t=i_t,\dots,X_0=i_0)=\Pr(X_{t+1}=j\mid ...
2
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0answers
53 views

Autocorrelation function of random process

Let $X_t$ be a wide sense stationary random process indexed by $t\in\mathbb{R}$ with finite mean and variance. (http://en.wikipedia.org/wiki/Stationary_process) Q1) Is the autocorrelation function ...
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2answers
41 views

Stochastic process vs Random process! [duplicate]

I am taking a course in stochastic process this time. As I read through a couple of books every one mentioned that stochastic process is also a random process. So, my confusion is why we call ...
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18 views

show the AR process is non-stationary. [closed]

A first order auto-regressive (AR) process $u(n)$ satisfies the equation: $$ u(n) + a_1 u(n-1) = v(n) $$ where $a$ is a constant and $v(n)$ is a white-noise process with variance $\sigma^2_v$. Show ...
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10 views

Ergodic Versus non-Ergodic Processes

Besides time averaging not carrying over to the ensemble average (in the limit), what are the pros and cons of ergodic and non-ergodic processes? Suppose you were in an engineering situation and you ...
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1answer
15 views

Transition from Variance to Covariance in the proof is unclear.

I'm looking at the proof (not fully presented here) and cannot understand one thing. How is transition from Variance to Covariance justified? I mean, isn't Variance equal to Covariance if this is ...
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1answer
47 views

Isometric in stochastic integral.

If $\{X_t\}_{t\ge 0}$ is a simple process. i.e.$0=t_0\le t_1\le\cdots\le t_n=T$ $\exists \xi_i\in\mathcal F_{t_i}$ s.t.$X_t(\omega)=\xi_i(\omega)$ when $t\in[t_i,t_{i+1}].$ $\{W_t\}_{t\ge 0}$ is a ...
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22 views

Mutual information between two Gaussian distribution

Suppose we have two variables $x_i$ and $x_j$ with covariance matrices $P_i$ and $P_j$ and cross-covariance $P_{ij}$. I'd like to find the mutual information on them. From reverse engineering of some ...
3
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1answer
49 views

My understanding of “$\sigma-$algebra represents information”.

In stochastic process $\{X_t\}_{t\ge0}$ adapted to $\{\mathcal F_t\}_{t\ge0}$ where $\mathcal F_s\subset\mathcal F_t,\forall s<t$. Many textbook say that $\{\mathcal F_t\}_{t\ge0}$ represents a ...
2
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1answer
20 views

Interpretation for the determinant of a stochastic matrix?

Is there a probabilistic interpretation for the determinant of a stochastic matrix (i.e. an $n \times n$ matrix whose columns sum to unity)?
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1answer
30 views

Prove of Stopping time

Let $(X_k)_{k\in\mathbb{N}}$ be iid random variables with $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$. Let $Z_n=\prod_{k=1}^n(1+X_k)$, so $Z_n$ a martingale. Consider ...
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29 views

Continuous time changes

Given a positive process $v_t$ with jumps, we define another process $T_t=\int_{0}^{t}v_sds$. I have troubles understanding why $T_t$ is absolutely continuous. Any references are highly appreciated ...
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1answer
14 views

Show that $\int _{{0}}^{{t}}h(s)dW_{{s}}=h(t)W_{{t}}-\int _{{0}}^{{t}}h^{{\prime}}(s)W_{{s}}ds\quad\mbox{ a.s.}$ [closed]

I have to show that $$\int _{{0}}^{{t}}h(s)dW_{{s}}=h(t)W_{{t}}-\int _{{0}}^{{t}}h^{{\prime}}(s)W_{{s}}ds\quad\mbox{ a.s.}$$ Can I have any tip how to start?
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1answer
38 views

Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
1
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0answers
10 views

Holder continuity, brwonian motion [duplicate]

Let $B$ stand for a brownian motion on a finite interval $[0,1]$. If i am not wrong, i think that there exists a positive constant $c$, such that almost surely, for h small enough , for all $0< t ...
0
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0answers
15 views

integrability condition stochastic process

Consider the finite time interval $[0,T]$ and the stochastic process $(X(t); t\leq s)$ Can the integral \begin{align} \int_{0}^{T}X(s)ds \end{align} de defined if the stochastic process $X$ is not ...
0
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1answer
19 views

Where do we encounter sequence of infinite events of which we would like to study probabilities?

I have come across sequence of functions and numbers in the context of approximation theory and understand that a lot of theory of functional analysis came out with the idea to approximate solutions ...
1
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1answer
47 views

Stopped sigma-algebra for a counting process

let $(\Omega, \mathcal{A}, P)$ be a probability space and $(N_t)_{t \geq 0}$ a right-continuous counting process with jumps of size 1, $N_0 = 0$ and canonical filtration $\mathcal{F}_t := \sigma( N_u ...
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0answers
25 views

Brownian motion-Holder [closed]

there exists a positive constant $c$, such that almost surely, for h small enough , for all $0< t < 1- h$ \begin{align} |B(t+h)-B(t)| < c\sqrt{h\log(1/h)} \end{align} As a result ...
2
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0answers
75 views
+50

numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ In this linear PDE: \begin{cases} B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...
0
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1answer
41 views

Problem with understading “mixed” integration

Using standard notation: $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t, \:\:X_0=x \tag{1}$$ Now in my script it is said that if we integrate both sides, we get: $$X_t=x+\int_0^t ...
0
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0answers
48 views

balls in bins — waiting time until $k$ bins are occupied

Consider the classic balls in bins problem: we throw balls one by one into $n$ bins independently and uniformly. Define $\tau(k)$ for $1 \le k \le n$ to be the number of balls we have thrown until $k$ ...
0
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0answers
33 views

Deriving Spectral density of White noise from Brownian motion

This is homework so no answers please Here is the problem and my answers (so please tell me if I made any mistakes): I am not asking you to compute the sum at the end, but to tell me if I made any ...
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42 views

Stochastically continuous but a.s. discontiuous process

This is a homework question so no answers please The problem is: Find a process $X_{t}$ s.t. $\forall t_{0}\geq 0$ and $\varepsilon>0$ we have $lim_{n\to ...
0
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1answer
20 views

Distribution of hitting position of line by brownian motion.

What is known about the distribution of the hitting position of a line by a 2d brownian motion? I've tried to make some simulations of a 2d brownian motion where every computational step has a ...
0
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1answer
54 views

Brownian motion - Hölder continuity

Let $B$ stand for a Brownian motion on a finite interval $[0,1]$. If I am not wrong, I think that there exists a positive constant $c$, such that almost surely, for $h$ small enough , for all $0< t ...
0
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0answers
18 views

Extension of martingale representation theorem.

It seems that the proof I am reading of the Martingale Representation Theorem, "A square integrable RCLL martingale which is adapted to the augmented filtration of a Brownian Motion must be an Ito ...
2
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1answer
39 views

Square integrable stochastic process

Suppose that for a stochastic process we have \begin{align} \mathbb{E}\left[\int_{0}^{T}X^{2}(t)dt \right]<\infty \end{align} where $T<\infty$. Does it holds that $|X(t)|<M$, where $M$ ...
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0answers
16 views

Dolphin in a pool - using Kolmogorov's forward equations

Problem A dolphin swims between 3 different pools, A B and C. The time is spend in each pool, before going to the next one, is Exp(1/2). The possible ways for it to travel is A to B. B to C. C ...
2
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1answer
28 views

Is this a Markov chain? [duplicate]

Let $\{\xi_n \}_{n \geq 1}$ be i.i.d random variables taking values on $\mathbb{Z}$. Let $\xi_0 = 0$. $S_n = \sum\limits_{i=1}^{n} \xi_i,$ where $S_0=0$ $Y_n = \sum\limits_{i=0}^{n} S_i$. My ...
0
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0answers
32 views

Integral of Squared Brownian Motion given the integral of a regular Brownian motion?

So I'm looking to calculate the variance of the following integral: $$\operatorname{Var}\left(\int_0^b W(s)^2\,ds \mid \int_0^b B^y(s)\,ds\right)$$ where $W(s)$ is a standard Brownian motion, and ...
1
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1answer
28 views

$\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum _{i=0}^{n}|B_{\frac{i+1}{n}}-B_{\frac{i}{n}}|^{p}=c_{p}$

This is a Homework question, so please do not answer it. Find real constants $a_{p},c_{p}$ s.t. $\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum ...
3
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1answer
31 views

Mean exit time / first passage time for a general symmetric Markov chain

Suppose I have a Markov chain as depicted in the following figure: where $N$ is even. State 0 and $N$ are the two sinks of the chain. The transition probabilities have the following property: ...